Question

How do you determine the base when solving for x by converting to a log?

The answer was supposed to be

x= ln(3/2) / 4ln(1.0125)

why is it ln?​

Answer 1

`$x=(\ln \left((3)/(2)\right))/(4 \ln (1.0125))`

Solution:

Given expression:

`$(1.0125)^(4 x)=(3)/(2)`

To solve the expression:

`$(1.0125)^(4 x)=(3)/(2)`

If f(x) = g(x) then ln(f(x)) = ln(g(x)).

Using the above condition, we can write

`$\ln \left(1.0125^(4 x)\right)=\ln \left((3)/(2)\right)`

Apply log rule:
`\log _(a)\left(x^(b)\right)=b \cdot \log _(a)(x)`

`$4 x \ln (1.0125)=\ln \left((3)/(2)\right)`

Divide both side of the equation by
`4 \ln (1.0125)`.

`$(4 x \ln (1.0125))/(4 \ln (1.0125))=(\ln \left((3)/(2)\right))/(4 \ln (1.0125))`

`$x=(\ln \left((3)/(2)\right))/(4 \ln (1.0125))`

The answer is
`x=(\ln \left((3)/(2)\right))/(4 \ln (1.0125))`.